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This third volume of the 31st annual proceedings of the International Group for the Psychology of Mathematics Education conference presents research reports for author surnames beginning Han- through Miy-. Reports include: (1) Elementary Education Students' Memories of Mathematics in Family Context (Markku S. Hannula, Raimo Kaasila, Erkki Pehkonen, and Anu Laine); (2) Mistake-Handling Activities in the Mathematics Classroom: Effects of an In-Service Teacher Training on Students' Performance in Geometry (Aiso Heinze and Kristina Reiss); (3) Gender Similarities instead of Gender Differences: Students' Competences in Reasoning and Proof (Aiso Heinze, Stefan Ufer, and Kristina Reiss); (4) Studying Lesson Structure from the Perspective of Students' Meaning Construction: The Case of Two Japanese Mathematics Classrooms (Keiko Hino); (5) A Framework for Creating or Analyzing Japanese Lessons from the Viewpoint of Mathematical Activities: A Fraction Lesson (Kenji Hiraoka and Kaori Yoshida-Miyauchi); (6) Revisiting Discourse as an Instructional Resource: Practices that Create Spaces for Learning and Student Contributions (Lynn Liao Hodge, Qing Zhao, Jana Visnovska, and Paul Cobb); (7) An Illustration of Students' Engagement with Mathematical Software Using Remote Observation (Anesa Hosein, James Aczel, Doug Clow, and John T. E. Richardson); (8) Geometric Calculations are More than Just the Application of Procedural Knowledge (Hui-Yu Hsu); (9) Constructing Pedagogical Representations to Teach Linear Relations in Chinese and U.S. Classrooms (Rongjin Huang and Jinfa Cai); (10) Teachers as Researchers: Putting Mathematics at the Core (Danielle Huillet); (11) Can You Convince Me: Learning to Use Mathematical Argumentation (Roberta Hunter); (12) On the Mathematical Knowledge under Construction in the Classroom: A Comparative Study (M. Kaldrimidou, H. Sakonidis, and M. Tzekaki); (13) Students' Beliefs and Attitudes about Studying and Learning Mathematics (Eleftherios Kapetanas and Theodosios Zachariades); (14) "How Can We Describe the Relation between the Factored Form and the Expanded Form of These Trinomials? We Don't even Know If Our Paper-and-Pencil Factorizations are Right": The Case for Computer Algebra Systems (CAS) with Weaker Algebra Students (Carolyn Kieran and Caroline Damboise); (15) What Is a Beautiful Problem? An Undergraduate Students' Perspective (Boris Koichu, Efim Katz, and Abraham Berman); (16) Can Lessons Be Replicated? (Angelika Kullberg); (17) Problem Posing as a Means for Developing Mathematical Knowledge of Prospective Teachers (Ilana Lavy and Atara Shriki); (18) Activity-Based Class: Dilemma and Compromise (KyungHwa Lee); (19) Induction, Analogy, and Imagery in Geometric Reasoning (KyungHwa Lee; MinJung Kim; GwiSoo Na, DaeHee Han, and SangHun Song); (20) The Analysis of Activity That Gifted Students Construct Definition of Regular Polyhedra (KyungHwa Lee, EunSung Ko, and SangHun Song); (21) Multiple Solution Tasks as a Magnifying Glass for Observation of Mathematical Creativity (Roza Leikin and Miri Lev); (22) Interactive Whiteboards as Mediating Tools for Teaching Mathematics: Rhetoric or Reality? (Steve Lerman and Robyn Zevenbergen); (23) From Construction to Proof: Explanations in Dynamic Geometry Environment (Allen Leung and Chi Ming Or); (24) Prospective Middle School Teachers' Knowledge in Mathematics and Pedagogy for Teaching--The Case of Fraction Division (Yeping Li and Dennie Smith); (25) Improving Students' Algebraic Thinking: The Case of Talia (Kien Lim); (26) The Effect of a Mentoring Development Program on Mentors' Conceptualizing Mathematics Teaching and Mentoring (Pi-Jen Lin); (27) Uses of Examples in Geometric Conjecturing (Miao-Ling Lin and Chao-Jung Wu); (28) Algebrification of Arithmetic: Developing Algebraic Structure Sense in the Context of Arithmetic (Drora Livneh and Liora Linchevski); (29) The Potential of Patterning Activities to Generalization (Hsiu-Lan Ma); (30) Infinite Magnitude vs. Infinite Representation: The Story of [Pi] (Ami Mamolo); (31) The Ability of Sixth Grade Students in Korea and Israel to Cope with Number Sense Tasks (Zvia Markovits and JeongSuk Pang); (32) Creating Your Own Symbols: Beginning Algebraic Thinking with Indigenous Students (Chris Matthews, Tom J. Cooper, and Annette R. Baturo); (33) Exploring Students' Mathematics-Related Self Image as Learners (Silvana Martins Melo and Marcia Maria Fusaro Pinto); (34) Difficulties on Understanding the Indefinite Integral (N. Metaxas); (35) Detecting the Emergence and Development of Mathematical Discourse: A Novel Approach (Christina Misailidou); and (36) The Nature and Role of Proof When Installing Theorems: The Perspective of Geometry Teachers (Takeshi Miyakawa and Patricio Herbst). (Individual papers contain references.) [For other volumes in the series, see ED499417, ED499418, and ED499419.]

This document contains the second volume of the proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Conference papers are centered around the theme of "Learners and Learning Environments." This volume features 43 research reports by presenters with last names beginning between Adl and Fre: (1) Working with Learners' Mathematics: Exploring a Key Element of Mathematical Knowledge for Teaching (Jill Adler, Zain Davis, Mercy Kazima, Diane Parker, and Lyn Webb); (2) A Comparison between Teachers' and Pupils' Tendency to Use a Representativeness Heuristic (Thekla Afantiti-Lamprianou, Julian S. Williams, and Iasonas Lamprianou); (3) Purposeful Task Design and the Emergence of Transparency (Janet G. Ainley, Liz Bills, and Kirsty Wilson); (4) A Developmental Model for Proportional Reasoning in Ratio Comparison Tasks (Silvia Alatorre and Olimpia Figueras); (5) Referential and Syntactic Approaches to Proof: Case Studies from a Transition Course (Lara Alcock and Keith Weber); (6) Teachers' Beliefs about Students' Development of the Pre-Algebraic Concept of Equation (Vassiliki Alexandrou-Leonidou and George N. Philippou); (7) Developing Students' Understanding of the Concept of Fractions as Numbers (Solange Amorim Amato); (8) Multiple Representations in 8th Grade Algebra Lessons: Are Learners Really Getting It? (Miriam Amit and Michael N. Fried); (9) Reform-Oriented Teaching Practices: A Survey of Primary School Teachers (Judy Anderson and Janette Bobis); (10) The Genesis of Signs by Gestures: The Case of Gustavo (Ferdinando Arzarello, Francesca Ferrara, Ornella Robutti, and Domingo Paola); (11) Students' Experience of Equivalence Relations: A Phenomenological Approach (Amir H. Asghari and David Tall); (12) How Series Problems Integrating Geometric and Arithmetic Schemes Influence Prospective Secondary Teachers' Pedagogical Understanding (Leslie Aspinwall, Kenneth L. Shaw, and Hasan Unal); (13) Dealing with Learning in Practice: Tools for Managing the Complexity of Teaching and Learning (Sikunder Ali Baber and Bettina Dahl); (14) Situations of Psychological Cognitive No-Growth (Roberto R. Baldino and Tania C. B. Cabral); (15) Good CAS Written Records: Insight from Teachers (Lynda Ball and Kaye Stacey); (16) Developing Procedure and Structure Sense of Arithmetic Expressions (Rakhi Banerjee and K. Subramaniam); (17) Struggling with Variables, Parameters, and Indeterminate Objects, or How to Go Insane in Mathematics (Caroline Bardini, Luis Radford, and Cristina Sabena); (18) Exploring How Power is Enacted in Small Groups (Mary Barnes); (19) A Framework for the Comparison of PME Research into Multilingual Mathematics Education in Different Sociolinguistic Settings (Richard Barwell); (20) Vygotsky's Theory of Concept Formation and Mathematics Education (Margot Berger); (21) Preservice Teachers' Understandings of Relational and Instrumental Understanding (Kim Beswick); (22) The Transformation of Mathematics in On-Line Courses (Marcelo C. Borba); (23) Using Cognitive and Situated Perspectives to Understand Teacher Interactions with Learner Errors (Karin Brodie); (24) Identification of Affordances of a Technology-Rich Teaching and Learning Environment (TRTLE) (Jill P. Brown); (25) The "A4-Project": Statistical World Views Expressed through Pictures (Michael Bulmer and Katrin Rolka); (26) A Whole-School Approach to Developing Mental Computation Strategies (Rosemary Callingham); (27) A Comparison of Perceived Parental Influence on Mathematics Learning among Students in China and Australia (Zhongjun Cao, Helen Forgasz, and Alan Bishop); (28) Using Word Problems in Malaysian Mathematics Education: Looking beneath the Surface (Kah Yein Chan and Judith Mousley); (29) Constructing Pedagogical Knowledge of Problem Solving: Preservice Mathematics Teachers (Olive Chapman); (30) Revisiting a Theoretical Model on Fractions: Implications for Teaching and Research (Charalambos Y. Charalambous and Demetra Pitta-Pantazi); (31) Students' Reflection on Their Sociomathematical Small-Group Interaction: A Case Study (Petros Chaviaris and Sonia Kafoussi); (32) Investigating Teachers' Responses to Student Misconceptions (Helen L. Chick and Monica K. Baker); (33) Studying the Distribution of Responsibility for the Generation of Knowledge in Mathematics Classrooms in Hong Kong, Melbourne, San Diego and Shanghai (David Clarke and Lay Hoon Seah); (34) Indigenous and Non-Indigenous Teaching Relationships in Three Mathematics Classrooms in Remote Queensland (Tom J. Cooper, Annette R. Baturo, and Elizabeth Warren); (35) Exploring the Strategies Used by Grade 1 to 3 Children through Visual Prompts, Symbols and Worded Problems: A Case for a Learning Pathway for Number (Ty Corvell Cranfield, Cally Kuhne, and Gary Powell); (36) Primary Students' Knowledge of the Properties of Spatially-Oriented Diagrams (Carmel Diezmann); (37) A Conceptual Framework for Studying Teacher Preparation: The Pirie-Kieren Model, Collective Understanding, and Metaphor (Maria A. Droujkova, Sarah B. Berenson, Kelli Slaten, and Sue Tombes); (38) Mathematical Modelling with 9-Year-Olds (Lyn English and James Watters); (39) Exploring "Lesson Study" in Teacher Preparation (Maria L. Fernandez); (40) Child-Initiated Mathematical Patterning in the Pre-Compulsory Years (Jillian Fox); (41) The Tacit-Explicit Nature of Students' Knowledge: A Case Study on Area Measurement (Cristina Frade); (42) Teachers as Interns in Informal Mathematics Research (John M. Francisco and Carolyn A. Maher); and (43) Exploring Excellence and Equity within Canadian Mathematics Classrooms (George Frempong). (Individual papers contain references.)